- #1

Keru

- 20

- 1

So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result:

W = 2a

_{z}

^{2}- a

_{x}

^{2}- a

_{y}

^{2}+ 2a

_{z}

^{+ 2}-a

_{x}

^{+ 2}-a

_{y}

^{+ 2}+ 2{a

_{z}, a

_{z}

^{+}}) - {a

_{x}, a

_{x}

^{+}}) - {a

_{y}, a

_{y}

^{+}}

As for the following part, I've proceeded like this, which I'm pretty sure it's wrong at some point.

In terms of the ladder operator, I'm using the basis:

|n

_{x}, n

_{y},n

_{z}>

With n

_{i}corresponding to the level of excitement on a certain coordinate. I have taken the first excited level to be:

|1,0,0>

I generally find the matrix elements like:

M

_{i,j}= <i|M^|j>

So my thought has been, I have to find them in the basis |1,0,0>, let's just:

<1,0,0|W|1,0,0>

Which looks terrible because that's clearly a 1x1 "matrix".

What am I missing? Do the matrix actually correspond to |1,0,0>, |0,1,0>, |0,0,1> since all these possible combinations correspond to the first energy state? Is that simply not the basis I should be using?

Any help would be greatly appreciated.